Problem 38 - Algebra and Trigonometry

Question

33–40 These problems involve distinguishable permutations. Room Assignments When seven students take a trip, they find a hotel with three rooms available—a room for one person, a room for two people, and a room for three people. In how many different ways can the students be assigned to these rooms? (One student has to sleep in the car.)

Step-by-Step Solution

Answer

There are 105 different ways to assign the students to the rooms.

Step 1: Determine Total Students and Required Distribution

There are 7 students in total. We need to assign 6 students into three rooms as follows: 1 room with 1 student, 1 room with 2 students, and 1 room with 3 students.

Step 2: Choose the Student for the Single Room

Select one student out of the 7 to be placed in the single room. This can be done in \( \binom{7}{1} = 7 \) ways.

Step 3: Choose Students for the Two-Person Room

After placing one student in the single room, there are 6 students left. Choose 2 students out of these 6 to be placed in the two-person room. This can be done in \( \binom{6}{2} = 15 \) ways.

Step 4: Assign Remaining Students to the Three-Person Room

There are now 4 students left. All 4 have to be placed in the three-person room. Since there are exactly 4 remaining, there is 1 way to make this assignment, \( \binom{4}{4} = 1 \).

Step 5: Calculate Total Permutations

Multiply the number of ways to assign each room to find the total permutations: \( 7 \times 15 \times 1 = 105 \).

Key Concepts

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination of objects. It helps us determine how to organize items in particular patterns under given conditions. In this exercise, we are applying combinatorics to find out how many distinct ways we can assign students to different rooms.

In our scenario, the focus is on grouping and organizing seven students into specific rooms with fixed capacities. The heart of combinatorics lies in understanding these arrangements and counting them accurately, which is achieved by using methods like permutations and combinations. By dividing the problem into smaller, more manageable parts, students learn to tackle complex organizational tasks methodically. Breaking down the problem, as shown in the original steps, is a testament to the power of combinatorics in solving real-world assignments like room allocation.

Permutation

Permutation refers to all the possible arrangements of a set of objects. In mathematics, it's used to determine the number of ways to order a group of items. However, distinguishable permutations specifically deal with arrangements where orders differ greatly depending on which items we choose.

In this room assignment problem, permutations help us choose and arrange students into the rooms efficiently. For instance, choosing 1 student for a single room out of 7 can be arranged in 7 different ways. Similarly, choosing 2 students from the remaining 6 for the two-person room results in 15 permutations. Finally, the remainder is simply slotted into the three-person room, which is done in 1 way. When we multiply these results together, we use the power of permutations to get a solution. Understanding permutations is crucial in scenarios where the order of selection matters, such as assigning students to rooms with varying capacities.

Room Assignment

Room assignment problems involve distributing a group of people into different spaces while considering room capacities and constraints. In our situation, there are 3 rooms and 7 students, with one student staying outside the hotel. Room assignment requires strategic thinking to utilize available resources effectively while respecting the constraints.

Each room in the problem has a specific capacity: one person, two people, and three people. The challenge is to assign students in such a way that each room's capacity is exactly met. This involves selecting the right number of students for each room, which is where the concepts of combinations and permutations come into play.

This type of problem is common in real-world scenarios where different spaces or resources are to be allocated efficiently. By practicing these kinds of problems, students learn not only mathematical techniques but also strategic planning skills that are useful in everyday situations.